Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
722 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 722 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #21043 | C2 | 1 | 4 | yes | maximal |
| 2 | #20799 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11364 | C2 | 1 | 4 | yes | maximal |
| 4 | #11212 | C2 | 1 | 2 | yes | maximal |
| 5 | #10893 | C2 | 1 | 2 | yes | maximal |
| 6 | #10819 | C2 | 1 | 2 | yes | maximal |
| 7 | C2×#427 | C22 | 1 | 4 | yes | — |
| 8 | #2905 | C22 | 1 | 2 | yes | — |
| 9 | #2873 | C22 | 1 | 2 | yes | — |
| 10 | C22×#97 | C22 | 1 | 1 | yes | characteristic |
| 11 | #2285 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2285 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1922 | — | 2 | 16 | no | — |
| 14 | #1847 | C4 | 1 | 4 | yes | complemented |
| 15 | #1847 | C4 | 1 | 4 | yes | complemented |
| 16 | #1740 | C22 | 1 | 2 | yes | — |
| 17 | #1740 | C22 | 1 | 2 | yes | — |
| 18 | C2×#341 | C22 | 1 | 2 | yes | — |
| 19 | #1735 | C22 | 1 | 2 | yes | — |
| 20 | #1733 | C22 | 1 | 4 | yes | — |
| 21 | #1733 | C22 | 1 | 4 | yes | — |
| 22 | C2×#339 | C22 | 1 | 4 | yes | — |
| 23 | #1729 | C22 | 1 | 4 | yes | — |
| 24 | C22×M16 | C23 | 1 | 4 | yes | — |
| 25 | C23×C8 | C23 | 1 | 2 | yes | — |
| 26 | C2×#117 | C23 | 1 | 4 | yes | — |
| 27 | C2×#117 | C23 | 1 | 2 | yes | — |
| 28 | C2×#117 | C23 | 1 | 2 | yes | — |
| 29 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 30 | #427 | — | 2 | 8 | no | — |
| 31 | #427 | — | 2 | 8 | no | — |
| 32 | C2×#107 | C2×C4 | 1 | 8 | yes | — |
| 33 | C2×#106 | C2×C4 | 1 | 8 | yes | — |
| 34 | #424 | — | 2 | 8 | no | — |
| 35 | #423 | — | 4 | 2 | no | — |
| 36 | #405 | — | 2 | 8 | no | — |
| 37 | C4×M16 | — | 4 | 2 | no | — |
| 38 | C2×#97 | — | 2 | 4 | no | — |
| 39 | C2×#97 | C2×C4 | 1 | 2 | yes | — |
| 40 | C2×#97 | C2×C4 | 1 | 2 | yes | — |
| 41 | #384 | C2×C4 | 1 | 2 | yes | — |
| 42 | #384 | C2×C4 | 1 | 2 | yes | — |
| 43 | #382 | C2×C4 | 1 | 2 | yes | — |
| 44 | #382 | C2×C4 | 1 | 2 | yes | — |
| 45 | #380 | — | 4 | 1 | no | — |
| 46 | #380 | — | 4 | 1 | no | — |
| 47 | C4×C4⋊C4 | — | 4 | 1 | no | — |
| 48 | C4×C4⋊C4 | — | 4 | 1 | no | — |
| 49 | #342 | — | 4 | 2 | no | — |
| 50 | #342 | — | 4 | 2 | no | — |
| 51 | #341 | D4 | 1 | 8 | yes | — |
| 52 | #340 | — | 2 | 4 | no | — |
| 53 | #339 | — | 2 | 8 | no | — |
| 54 | #339 | — | 2 | 8 | no | — |
| 55 | #339 | — | 2 | 8 | no | — |
| 56 | #336 | — | 2 | 8 | no | — |
| 57 | C23×C4 | — | 2 | 1 | no | — |
| 58 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 59 | C2×M16 | — | 2 | 4 | no | — |
| 60 | C2×M16 | — | 2 | 4 | no | — |
| 61 | C2×M16 | — | 2 | 4 | no | — |
| 62 | C2×M16 | — | 2 | 4 | no | — |
| 63 | C2×M16 | — | 2 | 4 | no | — |
| 64 | C2×M16 | C2×D4 | 1 | 8 | yes | — |
| 65 | C22×C8 | — | 2 | 4 | no | — |
| 66 | C22×C8 | — | 2 | 4 | no | — |
| 67 | C22×C8 | — | 2 | 4 | no | — |
| 68 | C22×C8 | C22×C4 | 1 | 4 | yes | — |
| 69 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 70 | C22×C8 | #41 | 1 | 4 | yes | — |
| 71 | #117 | — | 2 | 4 | no | — |
| 72 | #117 | — | 2 | 4 | no | — |
| 73 | #117 | — | 2 | 4 | no | — |
| 74 | #117 | — | 2 | 4 | no | — |
| 75 | #117 | — | 2 | 4 | no | — |
| 76 | #117 | — | 2 | 4 | no | — |
| 77 | #117 | — | 2 | 4 | no | — |
| 78 | #117 | — | 2 | 4 | no | — |
| 79 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 80 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 81 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 82 | C2×C4⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 83 | C2×C4⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 84 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 85 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 86 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 87 | C2×C42 | — | 2 | 1 | no | — |
| 88 | C2×C42 | — | 2 | 2 | no | — |
| 89 | C2×C42 | — | 2 | 2 | no | — |
| 90 | C2×C42 | — | 2 | 2 | no | — |
| 91 | C2×C42 | — | 2 | 4 | no | — |
| 92 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 94 | C2×C42 | #41 | 1 | 4 | yes | — |
| 95 | #107 | — | 2 | 16 | no | — |
| 96 | #106 | — | 2 | 16 | no | — |
| 97 | C4⋊C8 | — | 4 | 8 | no | — |
| 98 | #97 | — | 2 | 4 | no | — |
| 99 | #97 | — | 2 | 4 | no | — |
| 100 | #97 | — | 4 | 4 | no | — |
| 101 | C4×C8 | — | 4 | 4 | no | — |
| 102 | #95 | — | 4 | 2 | no | — |
| 103 | #95 | — | 4 | 2 | no | — |
| 104 | C24 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 105 | C22×C4 | — | 2 | 1 | no | — |
| 106 | C22×C4 | — | 2 | 1 | no | — |
| 107 | C22×C4 | — | 2 | 4 | no | — |
| 108 | C22×C4 | — | 2 | 4 | no | — |
| 109 | C22×C4 | — | 2 | 4 | no | — |
| 110 | C22×C4 | — | 4 | 4 | no | — |
| 111 | C22×C4 | — | 8 | 2 | no | — |
| 112 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 113 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 114 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 115 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 116 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 117 | C22×C4 | 2−1+4 | 1 | 2 | yes | — |
| 118 | C22×C4 | 2−1+4 | 1 | 4 | yes | — |
| 119 | M16 | — | 2 | 8 | no | — |
| 120 | M16 | — | 2 | 8 | no | — |
| 121 | M16 | — | 4 | 4 | no | — |
| 122 | M16 | — | 4 | 4 | no | — |
| 123 | C2×C8 | — | 2 | 4 | no | — |
| 124 | C2×C8 | — | 2 | 4 | no | — |
| 125 | C2×C8 | — | 2 | 8 | no | — |
| 126 | C2×C8 | — | 2 | 8 | no | — |
| 127 | C2×C8 | — | 2 | 8 | no | — |
| 128 | C2×C8 | — | 4 | 4 | no | — |
| 129 | C2×C8 | — | 4 | 4 | no | — |
| 130 | C2×C8 | C4×D4 | 1 | 8 | yes | — |
| 131 | C4⋊C4 | — | 2 | 8 | no | — |
| 132 | C4⋊C4 | — | 2 | 8 | no | — |
| 133 | C4⋊C4 | — | 4 | 4 | no | — |
| 134 | C4⋊C4 | — | 4 | 4 | no | — |
| 135 | C4⋊C4 | — | 8 | 4 | no | — |
| 136 | C4⋊C4 | — | 8 | 4 | no | — |
| 137 | C22⋊C4 | — | 4 | 4 | no | — |
| 138 | C22⋊C4 | — | 4 | 4 | no | — |
| 139 | C22⋊C4 | — | 4 | 4 | no | — |
| 140 | C22⋊C4 | — | 4 | 4 | no | — |
| 141 | C42 | — | 2 | 1 | no | — |
| 142 | C42 | — | 2 | 1 | no | — |
| 143 | C42 | — | 2 | 4 | no | — |
| 144 | C42 | — | 4 | 2 | no | — |
| 145 | C42 | — | 4 | 2 | no | — |
| 146 | C42 | — | 4 | 2 | no | — |
| 147 | C42 | — | 4 | 2 | no | — |
| 148 | C42 | — | 4 | 4 | no | — |
| 149 | C42 | — | 4 | 4 | no | — |
| 150 | C42 | #117 | 1 | 2 | yes | — |
| 151 | C42 | #117 | 1 | 2 | yes | — |
| 152 | C23 | — | 4 | 2 | no | — |
| 153 | C23 | #518 | 1 | 1 | yes | characteristic |
| 154 | C23 | #539 | 1 | 2 | yes | — |
| 155 | C23 | #550 | 1 | 2 | yes | — |
| 156 | C23 | #563 | 1 | 2 | yes | — |
| 157 | C2×C4 | — | 2 | 2 | no | — |
| 158 | C2×C4 | — | 2 | 2 | no | — |
| 159 | C2×C4 | — | 4 | 2 | no | — |
| 160 | C2×C4 | — | 4 | 2 | no | — |
| 161 | C2×C4 | — | 4 | 4 | no | — |
| 162 | C2×C4 | — | 4 | 4 | no | — |
| 163 | C2×C4 | — | 8 | 4 | no | — |
| 164 | C2×C4 | — | 8 | 4 | no | — |
| 165 | C2×C4 | — | 8 | 4 | no | — |
| 166 | C2×C4 | — | 8 | 8 | no | — |
| 167 | C2×C4 | — | 8 | 4 | no | — |
| 168 | C2×C4 | C2×#117 | 1 | 1 | yes | characteristic |
| 169 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 170 | C2×C4 | #519 | 1 | 1 | yes | characteristic |
| 171 | C2×C4 | #519 | 1 | 2 | yes | — |
| 172 | C2×C4 | #536 | 1 | 4 | yes | — |
| 173 | C2×C4 | #539 | 1 | 2 | yes | — |
| 174 | C2×C4 | #541 | 1 | 4 | yes | — |
| 175 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 176 | C2×C4 | #547 | 1 | 4 | yes | — |
| 177 | C2×C4 | #552 | 1 | 4 | yes | — |
| 178 | C2×C4 | #559 | 1 | 1 | yes | characteristic |
| 179 | C2×C4 | #563 | 1 | 2 | yes | — |
| 180 | C8 | — | 2 | 8 | no | — |
| 181 | C8 | — | 4 | 8 | no | — |
| 182 | C22 | — | 2 | 2 | no | — |
| 183 | C22 | — | 2 | 4 | no | — |
| 184 | C22 | — | 4 | 2 | no | — |
| 185 | C22 | — | 4 | 2 | no | — |
| 186 | C22 | #2313 | 1 | 1 | yes | characteristic |
| 187 | C22 | #2366 | 1 | 1 | yes | characteristic |
| 188 | C22 | #2381 | 1 | 1 | yes | characteristic |
| 189 | C22 | #2721 | 1 | 4 | yes | — |
| 190 | C4 | — | 4 | 2 | no | — |
| 191 | C4 | — | 4 | 2 | no | — |
| 192 | C4 | — | 8 | 4 | no | — |
| 193 | C4 | — | 8 | 4 | no | — |
| 194 | C4 | — | 16 | 4 | no | — |
| 195 | C4 | #2327 | 1 | 2 | yes | — |
| 196 | C4 | #2344 | 1 | 2 | yes | — |
| 197 | C4 | #2680 | 1 | 2 | yes | — |
| 198 | C4 | #2794 | 1 | 2 | yes | — |
| 199 | C2 | — | 2 | 1 | no | — |
| 200 | C2 | — | 2 | 4 | no | — |
| 201 | C2 | — | 2 | 1 | no | — |
| 202 | C2 | #14424 | 1 | 1 | yes | characteristic |
| 203 | C2 | #23197 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. #1847 | C4 | semidirect |
| 15. #1847 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21043 | 2 |
| #20799 | 2 |
| #11364 | 2 |
| #11212 | 2 |
| #10893 | 2 |
| #10819 | 2 |